We construct a sequence of concordance invariants for classical links, which depend on the peripheral isomorphism type of the nilpotent quotients of the link fundamental group. The terminology stems from the fact that we replace the Magnus expansion in the definition of Milnor's $\bar{\mu}$-invariants by the similar Campbell–Hausdorff expansion. The main point is that we introduce a new universal indeterminacy, which depends only on the number of components of the link. The Campbell–Hausdorff invariants are new, effectively computable and can efficiently distinguish (unordered and unoriented) isotopy types of links, as we indicate on several families of closed braid examples. They also satisfy certain natural dependence relations, which generalize well-known symmetries of the $\bar{\mu}$-invariants.

1991 Mathematics Subject Classification: 81S25, 46L10, 46L50, 47A60.

We investigate a class of triangular UHF algebras which have the special property that there exists a sequence of unital multiplicative contractive finite-rank conditional expectations of the algebra into itself, which converges strongly to the identity, whose ranges form an increasing chain with dense union. A partial classification is given in terms of a local finiteness property, and a delimiting counterexample is obtained. Associated inverse limit algebras are considered, and a characterization of residually finite dimensional triangular AF algebras is obtained.

1991 Mathematics Subject Classification: 47D25, 46J35.

The Novikov-Landweber algebra and the Steenrod algebra are set up in terms of the primitive differential operators $D_k=\sum_i x_i^{k+1}\frac{\partial}{\partial x_i}$ acting in the usual way on the integral polynomial ring ${\bf Z}[x_1,\ldots ,x_n,\ldots]$. A commutative wedge product $\vee$ for differential operators is introduced and it is shown that the iterated wedge product $D_k^{\vee r}$ is divisible by $r!$ as an integral operator. The divided differential operator algebra $\cal D$ is generated over the integers by the divided operators $\frac{D_k^{\vee r}}{r!}$ under the wedge product. $\cal D$ is additively isomorphic to the abelian group of symmetric functions in the variables $x_i$. Furthermore $\cal D$ is closed under composition of operators and admits a natural coproduct which makes it a Hopf algebra in two ways, with respect to the composition and wedge products. Under composition $\cal D$ is isomorphic to the Landweber-Novikov algebra. A Hopf sub-algebra is generated under composition by the integral Steenrod squares $SQ^r = \frac{D_1^{\vee r}}{r!}$ and reduces mod 2 to the Steenrod algebra. An explicit product formula for two wedge expressions is developed and used to derive Milnor's product formula for his basis elements in the Steenrod algebra. The hit problem in the Steenrod algebra is reformulated in terms of partial differential operators.

1991 Mathematics Subject Classification 55S10.

It is known that a Markov map $T$ of the unit interval preserves a measure $\mu$, say, equivalent to Lebesgue measure, and that almost every point of the interval has a forward orbit under $T$ that is uniformly distributed with respect to $\mu$. In the opposite direction the main result of this paper states that there is a set of points having Hausdorff dimension $1$ whose forward orbits are in a certain sense very far from being so distributed.

1991 Mathematics Subject Classification: 58F08, 28A80.